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Annals of Mathematics log t2/3 law of the two dimensional asymmetric simple exclusion process By Horng-Tzer Yau... log t2/3 law of the two dimensionalasymmetric simple exclusion process

Annals of Mathematics

(log t)2/3 law of the two dimensional asymmetric simple exclusion process

By Horng-Tzer Yau

(log t)2/3 law of the two dimensional

asymmetric simple exclusion process

By Horng-Tzer Yau*

Abstract

We prove that the diffusion coefficient for the two dimensional asymmetric

simple exclusion process with nearest-neighbor-jumps diverges as (log t) 2/3 tothe leading order The method applies to nearest and non-nearest neighborasymmetric simple exclusion processes

1 Introduction

The asymmetric simple exclusion process is a Markov process on{0, 1}Zd

with asymmetric jump rates There is at most one particle allowed per site and

thus the word exclusion The particle at a site x waits for an exponential time and then jumps to y with rate p(x − y) provided that the site is not occupied.

Otherwise the jump is suppressed and the process starts again The jumprate is assumed to be asymmetric so that in general there is net drift of thesystem The simplicity of the model has made it the default stochastic modelfor transport phenomena Furthermore, it is also a basic component for models[5], [12] with incompressible Navier-Stokes equations as the hydrodynamicalequation

The hydrodynamical limit of the asymmetric simple exclusion process wasproved by Rezakhanlou [13] to be a viscousless Burgers equation in the Eulerscaling limit If the system is in equilibrium, the Burgers equation is trivialand the system moves with a uniform velocity This uniform velocity can beremoved and the viscosity of the system, or the diffusion coefficient, can bedefined via the standard mean square displacement Although the diffusion

coefficient is expected to be finite for dimension d > 2, a rigorous proof was

obtained only a few years ago [9] by estimating the corresponding resolventequation Based on the mode coupling theory, Beijeren, Kutner and Spohn [3]

*Work partially supported by NSF grant DMS-0072098, DMS-0307295 and MacArthur fellowship.

conjectured that D(t) ∼ (log t) 2/3 in dimension d = 2 and D(t) ∼ t 1/3 in d = 1 The conjecture at d = 1 was also made by Kardar-Parisi-Zhang via the KPZ

equation

This problem has received much attention recently in the context of tegrable systems The main quantity analyzed is fluctuation of the current

in-across the origin in d = 1 with the jump restricted to the nearest right site,

the totally asymmetric simple exclusion process (TASEP) Consider the cial configuration that all sites to the left of the origin were occupied while allsites to the right of the origin were empty Johansson [6] observed that thecurrent across the origin with this special initial data can be mapped into alast passage percolation problem By analyzing resulting percolation problem

spe-asymptotically in the limit N → ∞, Johansson proved that the variance of the current is of order t 2/3 In the case of discrete time, Baik and Rains [2]analyzed an extended version of the last passage percolation problem and ob-

tained fluctuations of order t α , where α = 1/3 or α = 1/2 depending on the

parameters of the model Both the approaches of [6] and [2] are related to theearlier results of Baik-Deift-Johansson [1] on the distribution of the length ofthe longest increasing subsequence in random permutations

In [10] (see also [11]), Pr¨ahofer and Spohn succeeded in mapping thecurrent of the TASEP into a last passage percolation problem for a generalclass of initial data, including the equilibrium case considered in this article.For the discrete time case, the extended problem is closely related to the work[2], but the boundary conditions are different For continuous time, besides theboundary condition issue, one has to extend the result of [2] from the geometric

to the exponential distribution

To relate these results to our problem, we consider the asymmetric simpleexclusion process in equilibrium with a Bernoulli product measure of density

ρ as the invariant measure Define the time dependent correlation function in

equilibrium by

S(x, t) = η x (t); η0(0).

We shall choose ρ = 1/2 so that there is no net global drift,

x xS(x, t)

= 0 Otherwise a subtraction of the drift should be performed The diffusion

coefficient we consider is (up to a constant) the second moment of S(x, t):



x

x2S(x, t) ∼ D(t)t for large t On the other hand the variance of the current across the origin is

Therefore, Johansson’s result on the variance of the current can be interpreted

as the spreading of S(x, t) being of order t 2/3 The result of Johansson is for

special initial data and does not directly apply to the equilibrium case If wecombine the work of [10] and [2], neglect various issues discussed above, andextrapolate to the second moment, we obtain growth of the second moment as

t 4/3 , consistent with the conjectured D(t) ∼ t 1/3

We remark that the results based on integrable systems are not just forthe variance of the current across the origin, but also for its full limiting dis-tribution The main restrictions appear to be the rigid requirements of thefine details of the dynamics and the initial data Furthermore, it is not clearwhether the analysis on the current across the origin can be extended to the

diffusivity In particular, the divergence of D(t) as t → ∞ in d = 1 has not

been proved via this approach even for the TASEP

Recent work of [8] has taken a completely different approach It is based onthe analysis of the Green function of the dynamics One first used the duality

to map the resolvent equation into a system of infinitely-coupled equations.The hard core condition was proved to be of lower order Once the hard corecondition was removed, the Fourier transform became a very useful tool andthe Green function was estimated to degree three This yielded a lower bound

to the full Green function via a monotonicity inequality Thus one obtained

the lower bounds D(t) ≥ t 1/4 in d = 1 and D(t) ≥ (log t) 1/2 in d = 2 [8] In

this article, we shall estimate the Green function to degrees high enough to

determine the leading order behavior D(t) ∼ (log t) 2/3 in d = 2.

1.1 Definitions of the models Denote the configuration by η = (η x)x ∈Z d

where η x = 1 if the site x is occupied and η x = 0 otherwise Denote η x,y the configuration obtained from η by exchanging the occupation variables at

For two cylinder functions f , g and a density ρ, denote by f; g ρ the

covariance of f and g with respect to ν ρ:

f; g ρ = fg ρ − f ρ g ρ

Let P ρdenote the law of the asymmetric simple exclusion process starting from

the equilibrium measure ν ρ Expectation with respect to P ρ is denoted by E ρ.Let

is time independent and χ(ρ) = ρ(1 − ρ) in our setting.

The bulk diffusion coefficient is the variance of the position with respect

to the probability measure S ρ (x, t)χ −1 inZd divided by t; i.e.,

For simplicity, we shall restrict ourselves to the case where the jump is

symmetric in the y axis but totally asymmetric in the x axis; i.e., only the jump to the right is allowed on the x axis Our results hold for other jump

rates as well The generator of this process is given by

where we have combined the symmetric jump on the y axis into the last term.

We emphasize that the result and method in this paper apply to all asymmetricsimple exclusion processes; the special choice is made to simplify the notation.The velocity of the totally asymmetric simple exclusion process is explicitly

computed as v = 2(1 − 2ρ)e1 We further assume that the density is 1/2 so

that the velocity is zero for simplicity

Denote the instantaneous currents (i.e., the difference between the rate at

which a particle jumps from x to x + e i and the rate at which a particle jumps

from x + e i to x) by ˜ w x,x+e i:

˜

w x,x+e1 = η x[1− η x+e1], w˜x,x+e2 = η x − η x+e2

2(1.6)

We have the conservation law

Let w i (η) denote the renormalized current in the ith direction:

a finite number of terms in this sum vanish because ν ρ is a product measure

and g, h are mean zero From this inner product, we define the norm:

f 2=f, f.

(1.8)

Notice that all degree one functions vanish in this norm and we shall

identify the currents w with their degree two parts Therefore, for the rest of

this paper, we shall put

This is some variant of the Green-Kubo formula Since w2 = 0, D − I/2 is a

matrix with all entries zero except

ρ = 1/2 We believe that the method applies to general cases as well; see the

comment at the end of the next section for more details

Theorem 1.1.Consider the asymmetric simple exclusion process in d = 2 with generator given by (1.5) Suppose that the density ρ = 1/2 Then there exists a constant γ > 0 so that for sufficiently small λ > 0,

λ −2 | log λ| 2/3

e −γ| log log log λ|2 ≤

 ∞0

e −λt tD11 (t)dt ≤ λ −2 | log λ| 2/3

e γ | log log log λ|2.

From the definition, we can rewrite the diffusion coefficient as

dt

 t0

 s0

du

∞ u

dt e ưλ(tưu)

  t u

| log λ| 2/3 e ưγ| log log log λ|2 ≤ w1, (λ ư L) ư1 w1 ≤ | log λ| 2/3 e γ | log log log λ|2.

From the following well-known lemma, the upper bound holds without thetime integration For a proof, see [9]

Lemma 1.1 Suppose µ is an invariant measure of a process with ator L Then

gener-E µ



t ư1/2

 t0

w(η(s)) ds

2

≤ w1, (t ư1 ư L) ư1 w1.

(1.12)

Since w1 is the only non-vanishing current, we shall drop the subscript 1

2 Duality and removal of the hard core condition

Denote by C = C(ρ) the space of ν ρ-mean-zero-cylinder functions For afinite subset Λ ofZd , denote by ξΛ the mean zero cylinder function defined by

because h is assumed to be a cylinder function Denote by C n =∪1 ≤j≤n M j

the space of cylinder functions of degree less than or equal to n All mean zero cylinder functions h can be decomposed as a finite linear combination of

cylinder functions of finite degree : C = ∪ n ≥1 M n Let L = S + A where S is

the symmetric part and A is the asymmetric part Fix a function g in M n :

The asymmetric part A is decomposed into two pieces A = M + J so that M

maps M n into itself and J = J++ J − mapsM n intoM n −1 ∪ M n+1:

From now on, we shall refer to f (x1,· · · , x n) as a homogeneous function of

degree n vanishing on the complement of E1

With this identification, the coefficients of the current are given by

w1 (0, e1) = w1(e1, 0) := (w1){0,e }=−1/4

and zero otherwise Since we only have one nonvanishing current, we shalldrop the subscript 1 for the rest of this paper.

If g is a symmetric homogeneous function of degree n, we can check that

× [g(x1, · · · x i + σe β , · · · , x n)− g(x1, · · · , x i , , · · · , x n)]

where α is some constant and δ(0) = 1 and zero otherwise The constant α

is not important in this paper and we shall fix it so thatS is the same as the

discrete Laplacian with Neumann boundary condition on E1

The hard core condition makes various computations very complicated

In particular, the Fourier transform is difficult to apply However, if we areinterested only in the orders of magnitude, this condition was removed in [8]

We now summarize the main result in [8]

For a function F , we shall use the same symbol F  to denote the

where the discrete Laplacian is given by

Denote by π nthe projection onto functions with degrees less than or equal

to n Let L n be the projection of L onto the image of π n , i.e., L = π n Lπ n.The key result of [8] is the following lemma

Lemma 2.1 For any λ > 0 fixed, for k ≥ 1,

We have assumed that the process is given by (1.5); i.e., the jump is

sym-metric in the y axis and totally asymsym-metric in the x axis But the setup in this

section clearly applies to general jump rates as well The only difference is theanalysis of the equation (2.10) Since our main tool in the next few sections isthe Fourier transform, we expect it to be applicable to general translationallyinvariant jump rates and to yield similar results The more important assump-

tion for Theorem 1.1 is the density ρ = 1/2 For the current across the origin

in one dimension [2], [10], ρ = 1/2 is the only equilibrium density for which the

variance of the current across the origin is not the standard Gaussian For the

diffuseness the density ρ = 1/2 may not play such a critical role The reason

is that the operator M in (2.1) behaves like a drift operator In Fourier space,

it becomes a multiplication operator p1+· · · + p n Due to the average overthe translation, the relevant inner product (3.2) restricts the Fourier modes

to the hyperplane p1+· · · + p n = 0 Therefore, M essentially vanishes for all

densities with respect to the norm defined by the inner product (3.2) Morecareful analysis is still needed to determine if this heuristic argument is correct.For the current across the origin, on the other hand, there is no average over

translation and p1 behaves like an elliptic operator in d = 1 This explains its Gaussian behavior for ρ = 1/2.

3 Main estimate

We now introduce the following conventions: Denote the component of p

by (r, s) Denote p n = (p1,· · · , p n ), r n = (r1,· · · , r n) and sn = (s1,· · · , s n).The Fourier transform of

We can also compute the discrete Laplacian acting on F ,

In other words, when considering the inner product·, ·, we can consider

the class of ˆF (p1, · · · , p n) defined only on the subspace

j p j ≡ 0 mod 2π We

shall simply use the notation

j p j = 0 to denote the last condition

From now on, we work only on the moment space and all functions are

defined in terms of the momentum variables Let dµ n(pn) denote the measure

dµ n(pn) = 1

n! δ

n j=1

p j

n j=1

=−| log log λ|2ω(r n ), pn ∈ B τ

The main estimates of this paper are contained in the following theorem

Theorem 3.1 Let κ and τ be nonnegative numbers satisfying

Thus we can replace Ω in Theorem 3.1 by either V or U in the proof For the

rest of this paper, we shall follow the convention to denote the characteristic

function of a set A by A itself (instead of χ A)

Let V±,n+1 κ,2τ denote the positive and negative parts of Vκ,2τ n+1 Then

4.1 Decomposition into diagonal and off-diagonal terms Denote by Θ κ

To check the combinatorics, we notice that the total number of terms is

2

, the same as the total number of terms in (AF )2 The factors are obtained in

the following way Notice that in the formula of (AF )2 we have to choose two

indices We first fix the special two indices in one F to be, say, (1, 2) This gives a factor n(n + 1)/2 There is only one choice for the second index to be (1, 2) and this gives the first factor for the diagonal term There are 2(n − 1) choices to have either 1 or 2 and (n − 1)(n − 2)/2 choices to have neither 1

nor 2 These give the last two factors

Notice that by the Schwarz inequality, the off-diagonal term is bounded bythe diagonal term For the purposes of upper bound we only have to estimatethe diagonal term Since the number of the off-diagonal terms is bigger than

the diagonal terms by a factor of order n2, we have the upper bound



dµ n+1(pn+1)G 2τ

(pn+1)Θκ(pn+1)|A+F (p1, · · · , p n+1)|2(4.9)

≤ Cn4 F, K κ, G 2τ

n F  4.2 Preliminary remarks. Notice in the expression for K κ, G 2τ

Suppose at least one of|r n |, |r n+1 |, |s n |, |s n+1 | is not near 0 or π, say

After integrating p n − p n+1 , we change the variable u+ = p n + p n+1 to p n

Recall the normalization difference ((n + 1)!) −1 and (n!) −1 for dµ n+1 and dµ n.Thus,

negligible Therefore, we shall assume that

|r n |, |r n+1 |, |s n |, |s n+1 | ∈ [0, π/100] ∪ [99π/100, π].

(4.13)

We now divide the integration region according to |r n |, |r n+1 |, |s n |, |s n+1 |

in [0, π/100] or [99π/100, π] There are sixteen disjoint regions and the final

results are obtained by adding together the estimates from these sixteen joint regions For simplicity, we shall consider only the region that all these

dis-variables are in the interval [0, π/100] The estimates in all other regions are the same For example, suppose that r n+1 ∈ [99π/100, π] and the other three variables belong to [0, π/100] Let p n+1 = (π, 0) + ˜ p n+1 and define

This argument applies to all terms for the rest of this paper and we shall

from now on consider only this case The indices n, n + 1 are the two indices appearing in F (p1,· · · , p n −1 , p n + p n+1); they may change depending on thevariables we use in the future Notice that in this region,

ω(p j)∼ p2

j , j = n, n + 1, ω(p n ± p n+1)∼ (p n ± p n+1)2.

(4.15)

The main estimates of this paper are contained in the following theorem

Theorem...

We have assumed that the process is given by (1.5); i.e., the jump is

sym-metric in the y axis... class="text_page_counter">Trang 9

and zero otherwise Since we only have one nonvanishing current, we shalldrop the subscript for the rest of this